Angular Momentum Algebra. Abstract

Transcription

1 Angular Momentum Algebra Michael W. Kirson Department of Particle Physics Weizmann Institute of Science Rehovot, Israel (Dated: January 2002) Abstract This brief summary of the quantum theory of angular momentum is intended as a heuristic, reasonably self-contained presentation of useful results. It has no pretensions to rigour and certainly does not presume to supplant the existing detailed texts on the subject, which should be consulted for a more thorough treatment of the topics touched on below. A brief bibliography is given at the end. In addition, most texts on quantum mechanics have at least some dicussion of the quantum mechanics of angular momentum, some of them quite extensive. 1

2 I. BASIC DEFINITIONS In classical mechanics, the angular momentum of a point object is defined as the vector product of its position and momentum vectors, L = r p. In quantum mechanics, where r and p are operators, one for each component of each vector, this same definition produces a set of three operators, L x, L y and L z. From the standard commutation relation [r α, p β ] = i hδ αβ, where the subscripts α and β denote the indices x, y or z and δ αβ is the Krönecker delta, it follows that [L x, L y ] = i hl z (1.1) and similarly for any cyclic permutation of the indices x, y and z. These results generalise easily to collections of independent particles. Choosing h as the unit of angular momentum, and regarding these commutation relations as the defining characteristic of angular momentum, the general quantal definition of angular momentum will be taken to be as follows: Angular momentum is a physical observable represented by three hermitian operators j x, j y and j z which satisfy the commutation relations [j x, j y ] = ij z, and cyclic permutations. These operators are the components of a vector j. The change in notation from L to j is intended to indicate the possibility of generalisation of the concept of angular momentum beyond that associated with classical orbital motion. The operators of angular momentum generate an algebra (the commutator of any two operators in the set is a linear combination of operators from the same set). Since the significance of operators in quantum mechanics lies in their matrix elements, there is obvious interest in establishing the matrix representations of the angular momentum algebra in terms of standard basis functions. Though no two components of the angular momentum operator commute with one another, all three components compute with the quadratic form j 2 = jx 2 + j2 y + j2 z, and it may be established that this is the most general angular momentum operator with this property. According to the general principles of quantum mechanics, j 2 may be diagonalised simultaneously with any one component of j, and their eigenvalues may be used to label quantum states. The standard choice is to diagonalise j 2 and j z, though any other choice of the component to be diagonalised would be completely equivalent. 2

3 It turns out to be useful to look for what may be thought of as eigenoperators of j 2 and j z, namely operators Θ which satisfy [ j 2, Θ] = λθ and [j z, Θ] = mθ, where λ and m are numbers. Any linear combination of the operators j α satisfies the first relation, with λ = 0, but the second is less trivial. It is easily established that the only linear combinations of j α satisfying the second eigenoperator relation are j z itself, trivially, with m = 0, and the two operators j ± = j x ± ij y, with m = ±1. (Of course, any multiples of these operators will satisfy the same relationship, since it is linear.) These operators satisfy the commutation relations [ j 2, j ± ] = 0 (1.2) [j z, j ± ] = ±j ± (1.3) [j +, j ] = 2j z (1.4) Consider a set of states λm which are simultaneous eigenstates of j 2 and j z with eigenvalues λ and m, respectively, i.e. j 2 λm = λ λm (1.5) j z λm = m λm (1.6) Applying to these states the commutation relations for j ± one obtains j 2 j ± λm = [ j 2, j ± ] λm + j ± j 2 λm = λj ± λm and j z j ± λm = [j z, j ± ] λm + j ± j z λm = (m ± 1)j ± λm from which it follows that j ± λm = α (±) λm λ, m ± 1 (1.7) where α (±) λm is a numerical coefficient and it is assumed that the states λm are all normalised. The operators j ± are therefore referred to as step operators, changing the eigenvalue of j z by one unit, up or down respectively, while leaving the eigenvalue of j 2 unchanged. It should be noted that j ± are not hermitian operators, but that each is the hermitian conjugate of the other. 3

4 and The relations j + j = j 2 x + j2 y + i[j y, j x ] = j 2 j 2 z + j z (1.8) j j + = j 2 j 2 z j z (1.9) which follow from the definition of j ±, may be used to determine the coefficients α (±) λm, since α (±) λm 2 = j ± λm 2 = λm j j ± λm = λm j 2 j 2 z j z λm = λ m 2 m. As always, quantal states are defined, even when normalised, only up to an overall phase. There is thus complete freedom to choose the phases of the basis states λm in order to simplify the above results as far as possible. The standard choice, which can be shown always to be possible, is known as the Condon-Shortley phase convention and involves choosing the coefficients α (±) λm to be real and positive. So α (±) λm = λ m 2 m. (1.10) Since the norm of a quantal state must be non-negative, it follows that 0 λm j +j + + j j λm = λm j j + + j + j λm = λm 2 j 2 2jz λm 2 = 2(λ m 2 ) so that m 2 λ and, for any finite value of λ, there is a limitation on the possible values of m. Consider a specific state λm and apply to it the angular-momentum raising (step-up) operator j +. This produces a state λ, m + 1. Repeated applications of j + will produce a succession of states λ, m + n, where n is an arbitrary positive integer. For any initial values of λ and m, there will eventually be some value of n for which (m + n) 2 > λ, which is forbidden. Thus there must exist some maximum value of n, denoted n max, such that 4

5 j + λ, m + n max = 0. Then j j + λ, m + n max = [λ (m + n max ) 2 (m + n max )] λ, m + n max = 0 i.e. λ = (m + n max )(m + n max + 1). Similarly, repeated applications of the step-down operator j to the state λm will ultimately produce a state λ, m n such that (m n ) 2 > λ, which is again forbidden. There must thus also exist some maximum value of the positive integer n, denoted n min, such that j λ, m n min = 0. Then j + j λ, m n min = [λ (m n min ) 2 + (m n min )] λ, m n min = 0 i.e. λ = (m n min )(m n min 1). Therefore (m + n max )(m + n max + 1) = (m n min )(m n min 1), from which m + n max = (m n min ) and so 2m = n min n max. Recalling that n max and n min are non-negative integers, it follows that m must be an integer or half an odd integer (positive or negative), with a maximum allowed value and a minimum allowed value for any fixed value of λ. Denoting the maximum allowed value of m by j, the above results imply that the minimum allowed value is j and that λ = j(j + 1). To summarise, it is inherent in the angular momentum algebra that the simultaneous eigenstates of j 2 and j z are determined by the quantum numbers j and m, where j is a non-negative number, either an integer or half an odd integer, and m takes the 2j +1 values j to j in integer steps. The normalised states jm satisfy j 2 jm = j(j + 1) jm (1.11) j z jm = m jm (1.12) j ± jm = α (±) jm j, m ± 1 (1.13) α (±) jm = j(j + 1) m(m ± 1) (1.14) where the last equation embodies a phase convention for the states. For each j there is a (2j + 1)-dimensional matrix representation for the three angular momentum operators j x, j y and j z, where the rows and columns are labelled by the quantum number m. Since j ± (and hence j x and j y ) connect every state to a neighbouring state, there is no smaller subset 5

6 of linear combinations of the states jm, for given j, which is closed under the operation of all three operators j α (i.e. such that any member of the subset is converted into a linear combination only of members of the subset by any of the operators j α ). Such representations are said to be irreducible. Since the whole of the above discussion has been based only on the commutation relations of the operators, it holds true for any set of operators satisfying the same commutator algebra and, in particular, for the orbital angular momentum operators l = ( r p)/ h = i r. It is well known that the simultaneous eigenfunctions of l 2 and l z are the spherical harmonics Y lm (θ, φ), where l is restricted to be an integer in order for the function to be singlevalued over the range of its arguments. The Condon-Shortley phase convention implies Y lm (θ, φ) = ( )m Y l, m (θ, φ). A very useful general result can now be established concerning the matrix elements, in the angular momentum basis, of any operator which commutes with all the components of the angular momentum operator j. Denoting the operator by Θ, the condition [Θ, j α ] = 0 for α = x, y, z implies also that [Θ, j 2 ] = 0. Matrix elements of this equation may be taken between states ξjm, where ξ symbolises all other quantum numbers required to specify the states completely. These quantum numbers are irrelevant to the present discussion and will simply remain unchanged at each step of the argument. Thus 0 = ξjm [Θ, j 2 ] ξ j m = ξjm Θ j 2 j 2 Θ ξ j m = j (j + 1) ξjm Θ ξ j m j(j + 1) ξjm Θ ξ j m = [j (j + 1) j(j + 1)] ξjm Θ ξ j m where the hermiticity of j 2 has been used in operating to the left on the state ξjm and in taking real eigenvalues. It follows that ξjm Θ ξ j m = 0 unless j = j. (1.15) In precisely the same way, taking matrix elements of [Θ, j z ] = 0 leads to ξjm Θ ξ j m = 0 unless m = m. (1.16) 6

7 Finally, since [Θ, j + ] = 0, it follows that ξj, m + 1 Θ ξ j, m + 1 = ξj, m + 1 Θj + ξ jm /α (+) jm = ξj, m + 1 j + Θ ξ jm /α (+) jm = α j,m+1 ξjm Θ ξ ( ) jm /α (+) jm = ξjm Θ ξ jm, from the explicit form of α (±) jm. Therefore, any operator which commutes with all the components of j is diagonal in the quantum numbers j and m and its matrix elements in the basis ξjm are independent of m. In particular, the unit operator commutes with all components of j. Its matrix elements simply express the overlap between different states. The general result here then ensures that the overlap between any two eigenstates of a given angular momentum operator j, no matter what other quantum numbers are required to specify the states completely, will be zero if the states belong to different eigenvalues j or m and will be independent of m when the states belong to the same eigenvalues j and m. II. COMBINATION OF ANGULAR MOMENTA Consider a number of independent angular momenta, i.e. a number of triplets of operators j (a), where a is an index labelling a particular triplet, such that the components of different operators j (a) commute, [j (a) α, j (b) β ] = iδ abj (a) γ (2.1) where (α, β, γ) is a cyclic permutation of (x, y, z). These may be the orbital angular momenta of different particles, or the orbital and spin angular momentum of a single particle, or some more general combination. Defining the resultant or total angular momentum operator j by j α = a j (a) α, (2.2) it is easily checked that the three components of j do in fact satisfy the angular momentum algebra, hence justifying the nomenclature. There will thus be a set of eigenstates jm with all the properties established above. 7

8 In this case, it is easy to find additional operators which commute with j 2 and j z and with one another and whose eigenvalues can be used to supply additional quantum numbers characterising the total angular momentum eigenstates. In fact, any operator ( j (a) ) 2 commutes with all the components of any j (b) and hence with j 2, j z and any ( j (b) ) 2. It is therefore possible to diagonalise simultaneously all the ( j (a) ) 2 operators, together with j 2 and j z, and to produce the states j 1 j 2 j 3...j n jm, with j 2 j 1 j 2 j 3...j n jm = j(j + 1) j 1 j 2 j 3...j n jm (2.3) j z j 1 j 2 j 3...j n jm = m j 1 j 2 j 3...j n jm (2.4) ( j (a) ) 2 j 1 j 2 j 3...j n jm = j a (j a + 1) j 1 j 2 j 3...j n jm. (2.5) No single component j (a) α of the individual angular momenta commutes with j 2 and j z, because of the cross terms in j 2. However, partial sums of angular momenta, like ( j (a) + j (b) ) 2 or ( j (a) + j (b) + j (c) ) 2, etc., do commute with all the ( j (a) ) 2 and with j 2 and j z, so many more angular momentum quantum numbers can be simultaneously specified. Some caution is required, though, since two partial sums with overlapping sets of indices a, b,... will not commute with one another unless one is wholly contained as a sub-sum in the other. Clearly, several different sets of mutually commuting operators can be found, all including j 2 and j z and each defining a different basis of eigenvectors. If the operator j 2 is omitted, many more bases can be defined, the simplest being that in which ( j (a) ) 2 and j z (a), for all a, are diagonal, namely j 1 m 1 j 2 m 2 j 3 m 3...j n m n. Others would involve various partial sums of angular momenta, such as j 1 j 2 j 3 j 123 m 123 j 4 j 5 j 45 m 45...j n m n, for example. There is an embarrassing richness of alternatives, not all of which are really different from one another (they may differ simply by a relabeling of the j (a), for instance), and not all of which are of practical importance. In practice, detailed calculations seldom deal explicitly with more than four angular momenta at one time, and then the number of really different bases is relatively small and the relations between them quite easily established. It is both simplest and most instructive to tackle, in order, the case of two, then three and then four angular momenta, after which the generalisation to larger numbers is straightforward. In the case of two independent angular momenta j (1) and j (2), there are only two complete sets of mutually commuting angular momentum operators, the first being 8

9 ( j (1) ) 2, j z (1), ( j (2) ) 2, j z (2) and the second being ( j (1) ) 2, ( j (2) ) 2, j 2, j z, where j = j (1) + j (2). The corresponding eigenvectors are j 1 m 1 j 2 m 2 and j 1 j 2 ; jm, the former being referred to as uncoupled, the latter as coupled. For given values of the quantum numbers j 1 and j 2, these two sets of eigenvectors constitute different orthonormal bases for the representation of the angular momentum operators and must be connected by a unitary transformation. (In accordance with the general result obtained at the end of the preceding section, eigenvectors corresponding to different values of the quantum number j 1 or j 2 are orthogonal to one another.) The transformation coefficients between bases are called Clebsch-Gordan coefficients or vector-coupling coefficients and are most easily written down in Dirac notation: j 1 m 1 j 2 m 2 = jm j 1 j 2 ; jm j 1 m 1 j 2 m 2 j 1 j 2 ; jm (2.6) j 1 j 2 ; jm = m 1 m 2 j 1 m 1 j 2 m 2 j 1 j 2 ; jm j 1 m 1 j 2 m 2 (2.7) where j 1 m 1 j 2 m 2 j 1 j 2 ; jm = j 1 j 2 ; jm j 1 m 1 j 2 m 2 and the unitarity conditions are j 1 j 2 ; j m j 1 j 2 m 1 m 2 j 1 j 2 m 1 m 2 j 1 j 2 ; jm = δ jj δ mm (2.8) m 1 m 2 j 1 m 1j 2 m 2 j 1 j 2 ; jm j 1 j 2 ; jm j 1 m 1 j 2 m 2 = δ m1 m δ 1 m 2 m. (2.9) 2 jm For given j 1 and j 2, the quantum numbers m 1 and m 2 have their standard ranges (from j i to j i in unit steps) and for given j, the quantum number m has its standard range, but it has not yet been determined what are the possible values of j. For this, it is necessary to look into some of the detailed properties of the vector coupling process. Intuitively, since j z = j z (1) + j z (2), it is reasonable to expect m = m 1 + m 2. Therefore, the largest possible value attainable by m is m max = j 1 + j 2. It follows that the largest value attainable by j is also j 1 + j 2, since if there were a larger value of j it would be associated with larger values of m. There is only one basis function j 1 m 1 j 2 m 2 with m 1 +m 2 = j 1 +j 2, so that, necessarily, j 1 j 2 ; j 1 +j 2, j 1 +j 2 = j 1 j 1 j 2 j 2 and the corresponding Clebsch-Gordan coefficient is unity. (Note the off-hand introduction of a further phase convention.) It can be checked by straightforward, if lengthy, calculation that the uncoupled state j 1 j 1 j 2 j 2 is an eigenstate of j 2 and of j z, with quantum numbers j 1 + j 2 and j 1 + j 2. The state j 1 j 2 ; j 1 +j 2, j 1 +j 2 1 is now produced by acting on this extreme state with the step-down operator j. Since the value m = j 1 +j 2 1 can be produced either by m 1 = j 1, m 2 = j 2 1 9

10 or by m 1 = j 1 1, m 2 = j 2, there are two independent linear combinations of uncoupled states with this m value. One of them will be produced by j j 1 j 2 ; j 1 + j 2, j 1 + j 2, and the orthogonal linear combination must then be the state j 1 j 2 ; j 1 + j 2 1, j 1 + j 2 1 (with once again a free choice of phase). This procedure may now be repeated m = j 1 +j 2 2 can be produced in three different ways: m 1 = j 1, m 2 = j 2 2; m 1 = j 1 1, m 2 = j 2 1; m 1 = j 1 2, m 2 = j 2 ; so there are three independent linear combinations of uncoupled states with this m value. Two of them are produced by operating with the step-down operator j on the two coupled states found in the preceding steps, and are j 1 j 2 ; j 1 +j 2, j 1 +j 2 2 and j 1 j 2 ; j 1 +j 2 1, j 1 +j 2 2. The third linear combination, orthogonal to both of these, is then necessarily j 1 j 2 ; j 1 +j 2 2, j 1 +j 2 2. Continuing in this way, the coupled states j 1 j 2 ; jm may be produced in stepwise fashion, each step automatically determining the corresponding Clebsch-Gordan coefficients (and requiring one further phase choice, which can obviously be used to make all the Clebsch- Gordan coefficients real, since the Condon-Shortley convention makes all the α (±) jm coefficients real). The above process produces one extra state at each step, with a j value decreased by unity from that of the previous step, and will terminate when all the uncoupled basis states have been used up, which happens when either m 1 or m 2 reaches the lower end of its range. At this point, the number of different ways of making the appropriate value of m is equal to the number of different j values produced in the preceding steps. At the end of the process, all (2j 1 + 1)(2j 2 + 1) uncoupled basis functions j 1 m 1 j 2 m 2 will have been used to produce an equal number of coupled basis functions j 1 j 2 ; jm : j 1 +j 2 (2j 1 + 1)(2j 2 + 1) = (2j + 1) j=j min = (j 1 + j 2 + 1) 2 j 2 min from which j min = j 1 j 2. Thus the two angular momenta j 1 and j 2 may be coupled to a total angular momentum j 1 j 2, j 1 j 2 + 1, j 1 j 2 + 2,...j 1 + j 2 1, j 1 + j 2, each j value occurring once. The same results may be obtained more formally in the following fashion. Applying j z to the defining equation j 1 j 2 ; jm = m 1 m 2 j 1 m 1 j 2 m 2 j 1 j 2 ; jm j 1 m 1 j 2 m 2 10

11 produces m j 1 j 2 ; jm = m 1 m 2 j 1 m 1 j 2 m 2 j 1 j 2 ; jm (m 1 + m 2 ) j 1 m 1 j 2 m 2. On substituting the defining equation again in the left hand side, this may be rearranged into the form m 1 m 2 (m m 1 m 2 ) j 1 m 1 j 2 m 2 j 1 j 2 ; jm j 1 m 1 j 2 m 2 = 0. The states being summed over are mutually orthogonal and hence linearly independent, so the sum can vanish only if the coefficient of every state in the sum vanishes, i.e. j 1 m 1 j 2 m 2 j 1 j 2 ; jm = 0 unless m = m 1 + m 2. Thus the additive condition on m assumed intuitively above arises as a selection rule on the Clebsch-Gordan coefficients. (Note that the state j 1 m 1 j 2 m 2 is an eigenstate of j z with eigenvalue m 1 + m 2, while j 1 j 2 ; jm is an eigenstate of j z with eigenvalue m. The Clebsch-Gordan coefficient is an overlap between these two states, so vanishes unless their j z eigenvalues are equal. This is an alternative proof of the m selection rule.) The same technique may be used to produce a pair of recursion relations for the Clebsch- Gordan coefficients, by applying the operators j ± to the defining equation, substituting the defining equation again on the left hand side of the result, rearranging and using the linear independence of the uncoupled basis functions. The properties of the coefficients α (±) jm and of the basis vectors j i m i at the extremes of the range of m are important in ensuring that the ranges of summation on the two sides of the equation are compatible. The result is α (±) jm j 1 m 1 j 2 m 2 j 1 j 2 ; j, m ± 1 = α (±) j 1,m 1 1 j 1, m 1 1, j 2 m 2 j 1 j 2 ; jm +α (±) j 2,m 2 1 j 1 m 1 j 2, m 2 1 j 1 j 2 ; jm. (2.10) These recursion relations can be used to express all the Clebsch-Gordan coefficients for given j 1 and j 2 in terms of either j 1 j 1 j 2, j j 1 j 1 j 2 ; jj or j 1, j j 2, j 2 j 2 j 1 j 2 ; jj. (Consider, for example, the case m = j, with the upper sign, for which the left hand side vanishes and the right hand side, with the aid of the m selection rule, produces a recursion relation in m 1 alone.) The magnitude of the single remaining undetermined coefficient can be found by applying the unitarity condition in the form m 1 m 2 j 1 m 1 j 2 m 2 j 1 j 2 ; jm 2 = 1, and its phase can be chosen arbitrarily. For convenience, the phase is chosen so that the relevant coefficient is real and positive, j 1 j 1 j 2, j j 1 j 1 j 2 ; jj > 0, and the recursion relation then 11

12 ensures that all the Clebsch-Gordan coefficients are real. Therefore, j 1 m 1 j 2 m 2 j 1 j 2 ; jm = j 1 j 2 ; jm j 1 m 1 j 2 m 2 and only one form of the coefficient will be used henceforth. Since the quantum numbers j 1 and j 2 remain the same on both sides of the coefficient, the abbreviated form j 1 m 1 j 2 m 2 jm is sufficient and will be adopted from now on. Given the limitations on the projection quantum numbers m, it is clearly necessary that j 2 j j 1 j 2, or else all the Clebsch-Gordan coefficients vanish. Since the whole exercise could equally well have been carried out in terms of j 1, j j 2, j 2 j 2 j 1 j 2 ; jj (or, for that matter, in terms of the two alternative forms with m 1 = j 1 and m 2 = j 2 ), it is also necessary that j 1 j j 2 j 1. Combining the two sets of restrictions, it is seen that the values of j must satisfy the triangle inequality j 1 j 2 j j 1 + j 2. (2.11) From the general requirement that j be an integer or half an odd integer, and that m have the same character as j, and from the selection rule m = m 1 +m 2, it follows that j will be an integer if j 1 and j 2 are both integers or both half odd integers, and will be half an odd integer if only one of j 1 and j 2 is an integer. It is also worth noting that the whole treatment of the coupling of j 1 and j 2 was completely symmetric between the two component angular momenta until the choice of the phase of the determining coefficient in the recursion relation. The choice j 1 j 1 j 2, j j 1 j 1 j 2 ; jj > 0, rather than j 1, j j 2, j 2 j 2 j 1 j 2 ; jj > 0, introduces an asymmetry between j 1 and j 2, so that the coefficients j 1 m 1 j 2 m 2 jm and j 2 m 2 j 1 m 1 jm can differ at most by a sign. Similarly, since the recursion relation could have been used to relate all Clebsch-Gordan coefficients to those with m = j, instead of m = j, there will be a possible sign difference between coefficients with m 1, m 2, m or with m 1, m 2, m. Detailed discussion of the recursion relation, or inspection of the complicated explicit formula for the Clebsch-Gordan coefficients derived from group theory by various authors and quoted in the references, allow the extraction of the following symmetry properties of the coefficients: j 1 m 1 j 2 m 2 jm = ( ) j 1+j 2 j j 2 m 2 j 1 m 1 jm (2.12) = ( ) j 1+j 2 j j 1, m 1, j 2, m 2 j, m (2.13) = ( ) j 2+m 2 (2j + 1)/(2j 1 + 1) j 2, m 2 jm j 1 m 1 (2.14) = ( ) j 1 m 1 (2j + 1)/(2j 2 + 1) jmj 1, m 1 j 2 m 2. (2.15) 12

13 These properties suggest a basic symmetry among the three angular momenta involved the two constituent angular momenta j 1 and j 2 and the total angular momentum j. The same underlying symmetry is indicated by the triangle inequality satisfied by j 1, j 2 and j, which is equally valid for any division of the three angular momenta into two constitutents and a resultant. This symmetry is effectively exploited in an alternative form of the vectorcoupling coefficients, the Wigner 3-j symbol, defined by j 1 j 2 j 3 = ( ) j 1 j 2 m 3 j 1 m 1 j 2 m 2 j 3, m 3 / m 1 m 2 m 3 2j (2.16) The 3-j symbol vanishes unless j 1, j 2, j 3 satisfy the triangle inequality and m 1 +m 2 +m 3 = 0. It is left unchanged in value by any cyclic permutation of its three columns, but is multiplied by the phase factor ( ) j 1+j 2 +j 3 if any two columns are interchanged or if the signs of all three projection quantum numbers m i are reversed. Thus j 1 j 2 j 3 = 0 unless j 1 + j 2 + j 3 is even. The symbol also has the useful property that j 1 j 2 j 3 j 1 m 1 j 2 m 2 j 3 m 3 m 1 m 2 m 3 m 1 m 2 m 3 represents a state of three angular momenta coupled to total angular momentum zero which is symmetric under cyclic permutation of the three angular momenta. Some consequences of the angular-momentum coupling formalism are of general interest. In the case j = 0, the recursion relation for the Clebsch-Gordan coefficients becomes 0 = α (+) j 1,m 1 1 j 1, m 1 1, j 2, m α (+) j 2 m 2 j 1 m 1 j 2 m From the selection rules on the coefficients, necessarily j 1 = j 2 and m 1 = m 2. But from the explicit expression for the α (±) jm coefficients, α (+) j,m 1 = α (+) j, m, so the result reduces to jmj, m 00 = j, m 1, j, (m 1) 00, from which it follows that jmj, m 00 = ( ) j m jjj, j 00. But m jmj, m 00 2 = 1, so that jjj, j 00 = 1/ 2j + 1, where the standard phase convention has been used to select the positive square root. Finally, j 1 m 1 j 2 m 2 00 = ( ) j 1 m 1 δ j1 j 2 δ m1, m 2 / 2j (2.17) Using the symmetry relation of the Clebsch-Gordan coefficient, this may be rewritten in the form j 1 m 1 00 j 2 m 2 = δ j1 j 2 δ m1 m 2, which is self-evident. Alternatively, j 1 j 2 0 = ( ) j 1 m 1 δ j1 j 2 δ m1, m 2 / 2j (2.18) m 1 m

14 Another special case of the Clebsch-Gordan coefficient which is frequently useful can be obtained straightforwardly from the explicit formula for the coefficient, alluded to previously, namely jjj, j J0 = (2j)! (2J + 1)/(2j + J + 1)!(2j J)!. (2.19) Two identical systems, defined by their angular momentum j and by additional quantum numbers collectively denoted by ξ, can be coupled to a state of well-defined total angular momentum J as follows: (ξj) 2 JM = m 1 m 2 jm 1 jm 2 JM ξjm 1 ξjm 2. The dummy indices m i have the same range and may be exchanged to produce (ξj) 2 JM = m 1 m 2 jm 2 jm 1 JM ξjm 2 ξjm 1 = ( ) j+j J m 1 m 2 jm 1 jm 2 JM ξjm 2 ξjm 1, by the symmetry property of the Clebsch-Gordan coefficient. This may be rewritten (ξj) 2 JM = ( ) 2j J P 12 m 1 m 2 jm 1 jm 2 JM ξjm 1 ξjm 2 where P 12 is a permutation operator which exchanges the two systems in any state on which it acts (but has no effect on numerical coefficients like the Clebsch-Gordan coefficients). This final result states that P 12 (ξj) 2 JM = ( ) 2j J (ξj) 2 JM, (2.20) so that the coupled state (ξj) 2 JM is automatically symmetric or antisymmetric under interchange of the two systems, according as 2j J is even or odd, respectively. (Note the importance of the identity of the systems. For ξ 1 ξ 2, this result would not hold.) Considering that identical bosons, which have integer angular momentum, are required to form totally symmetric states, while identical fermions, which have half-odd-integer angular momentum, are required to form totally antisymmetric states, it follows that a pair of identical fermions or a pair of identical bosons can couple only to even total angular momentum J. This argument can be extended, in the case where each system is characterised by two or more independent angular momenta, to produce a correspondence between the sign of the phase factor ( ) J 1+J 2 + +J n and the symmetry of the state (ξj 1 j 2...j n ) 2 J 1 M 1 J 2 M 2...J n M n. Thus for systems characterised by angular momentum 14

15 and isospin, for example, selection rules arise for J + T, where T is the total isospin of the pair of identical systems. III. RECOUPLING OF ANGULAR MOMENTA As indicated at the beginning of the previous section, there will generally be more than two different bases applicable to problems involving more than two angular momenta. In the case of three angular momenta, the bases of interest are the totally uncoupled basis j 1 m 1 j 2 m 2 j 3 m 3 and the totally coupled basis j 1 j 2 j 3 jm, where the latter is not yet completely specified, being determined by the eigenvalues of only five, rather than six, mutually commuting operators. The totally coupled basis may be fully specified by defining one intermediate coupling, which can be done in three different ways (j 1 j 2 )j 12 j 3 jm, (j 1 j 3 )j 13 j 2 jm or j 1 (j 2 j 3 )j 23 jm, where the notation should be self-explanatory. The transition from the fully uncoupled to the fully coupled bases goes through the intermediate partially coupled basis, with Clebsch-Gordan coefficients being used to couple pairs of angular momenta at each step. So (j 1 j 2 )j 12 j 3 jm = j 12 m 12 j 3 m 3 j 12 m (j 1 j 2 )j 12 m 12 j 3 m 3 (3.1) m 12 m 3 = j 12 m 12 j 3 m 3 jm j 1 m 1 j 2 m 2 j 12 m 12 j 1 m 1 j 2 m 2 j 3 m 3, (3.2) m 1 m 2 m 3 m 12 and similarly for each of the other possible intermediate couplings. However, the existence of different bases does not imply a proliferation of states any one of the three fully coupled bases is equally valid, so there must exist unitary transformations between them. The vectors belonging to a particular basis, such as (j 1 j 2 )j 12 j 3 jm, are distinguished by the values of the six quantum numbers listed and are all mutually orthogonal, according to the general result proved at the end of section 1. However, vectors from different bases are not necessarily orthogonal, provided they have the same values of the common quantum numbers j 1, j 2, j 3, j and m. The overlaps between functions belonging to different bases are the coefficients of the unitary transformation between the bases, as is clear in Dirac notation: (j 1 j 2 )j 12 j 3 jm = j 1 (j 2 j 3 )j 23 (j 1 j 2 )j 12 j 3 j j 1 (j 2 j 3 )j 23 jm. (3.3) j 23 The transformation coefficient, again by the general result proved at the end of section 1, 15

16 is diagonal in j and m and independent of the value of m, which is thus dropped from the symbol for the overlap. Using the decomposition of the totally coupled states into totally uncoupled states in terms of Clebsch-Gordan coefficients, the transformation coefficient is explicitly written as j 1 (j 2 j 3 )j 23 j (j 1 j 2 )j 12 j 3 j = m 1 m 2 m 3 m 12 m 23 j 1 m 1 j 2 m 2 j 12 m 12 j 12 m 12 j 3 m 3 jm j 1 m 1 j 23 m 23 jm j 2 m 2 j 3 m 3 j 23 m 23. (3.4) Since the Clebsch-Gordan coefficients are real in the standard phase convention, so are these transformation (or recoupling) coefficients. Note that the recoupling coefficients are determined purely by the Clebsch-Gordan coefficients and are independent of any property of the states considered except their angular momentum quantum numbers. The recoupling coefficient is commonly written in terms of a more symmetrical quantity, the Wigner 6-j symbol, as follows: (j 1 j 2 )j 12 j 3 j j 1 (j 2 j 3 )j 23 j = ( ) j 1+j 2 +j 3 j +j 1 j 2 j 12 (2j )(2j ) j 3 j j 23. (3.5) An alternative notation is in terms of the Racah coefficient W(j 1 j 2 jj 3 ; j 12 j 23 ) = ( ) j 1+j 2 +j 3 j +j 1 j 2 j 12 j 3 j j 23. (3.6) The 6-j symbol is real and its selection rules are the four triangle inequalities, (j 1 j 2 j 12 ), (j 12 j 3 j), (j 2 j 3 j 23 ) and (j 1 j 23 j), where (j 1 j 2 j 3 ) j 1 j 2 j 3 j 1 + j 2. (3.7) Its symmetry properties may be read off from those of the Clebsch-Gordan coefficients (which is much more easily done when the latter are replaced by 3-j symbols). It turns out to be invariant under those permutations of its six arguments which leave the set of four triangle inequalities unchanged. Depicting these inequalities in the self-explanatory form it is quickly seen that this set of four pictures, and hence the 6-j symbol, is unchanged under arbitrary permutation of its three columns or under simultaneous inversion a b b a of any 16

17 pair of columns. Since the 6-j symbol is entirely independent of any projection quantum numbers m, it may be regarded as a rotational invariant. Other transformation coefficients can be expressed in terms of the one explicitly investigated above by suitable reorderings of the angular momenta being coupled, using the symmetries of the Clebsch-Gordan coefficients. For instance, (j 1 j 3 )j 13 j 2 j (j 1 j 2 )j 12 j 3 j = ( ) j 13+j 2 j j 2 (j 1 j 3 )j 13 j (j 1 j 2 )j 12 j = ( ) j 13+j 2 j ( ) j 1+j 2 j 12 j 2 (j 1 j 3 )j 13 j (j 2 j 1 )j 12 j 3 j = ( ) j 12+j 13 j 1 j ( ) j 2+j 1 +j 3 +j j 2 j 1 j 12 (2j )(2j ) j 3 j j 13 = ( ) j 2+j 3 +j 12 +j 13 j 2 j 1 j 12 (2j )(2j ) j 3 j j 13. (It should be noted that the 6-j symbol is defined by an overlap in which the three constituent angular momenta appear in precisely the same order in both totally coupled states. The reality of the 6-j symbol makes the order of the two totally coupled states in the overlap unimportant.) In manipulating phase factors, as has been done above, the following observations are useful. When j 1 and j 2 are coupled to produce j 3, any of the three may be, in principle, an integer or a half-odd-integer. However, as remarked above, in all cases either all three or only one of the j s must be integral. This is sufficient to ensure that the combinations j 1 + j 2 ± j 3 are integers, so that ( ) j 1+j 2 j 3, for instance, is real and is equal to its inverse, ( ) j 1 j 2 +j 3. However, ( ) j 1+j 2 j 3 is not necessarily equal to ( ) j 1+j 2 +j 3 these two phases are equal if j 3 is an integer, but opposite if it is a half-odd-integer. Similar arguments hold for j 1, j 2 and j 3 coupled to produce j, where j 1 + j 2 + j 3 + j is always an integer and ( ) j 1+j 2 +j 3 +j = ( ) j 1 j 2 j 3 j is real. Also, for any angular momentum j and its projection m, ( ) 4j = 1 = ( ) 4m and ( ) j m = ( ) m j is real, as is ( ) j+m = ( ) j m. From the unitarity of the transformation between bases and the definition of the 6-j symbol, the following useful relationships may be written down. j 1 j 2 j 12 j 1 j 2 j 12 (2j ) j 23 j 3 j j 23 j 3 j j 23 = δ j 12 j 12 /(2j ) (3.8) 17

18 j 1 m 1 j 2 m 2 j 12 m 12 j 12 m 12 j 3 m 3 jm = ( ) j 1+j 2+j 3+j j 1 j 2 j 12 (2j )(2j ) m 12 j 23 m 23 j 3 j j 23 j 1 m 1 j 23 m 23 jm j 2 m 2 j 3 m 3 j 23 m 23 (3.9) j 1 m 1 j 2 m 2 j 12 m 12 j 12 m 12 j 3 m 3 jm j 2 m 2 j 3 m 3 j 23 m 23 m 2 m 3 m 12 = ( ) j 1+j 2 +j 3 j +j 1 j 2 j 12 (2j )(2j ) j 3 j j 23 j 1m 1 j 23 m 23 jm (3.10) where the second equation is simply a rewriting of the definition of the unitary transformation coefficient and the third is derived from it by use of the orthogonality of the Clebsch-Gordan coefficients. A sum rule for the 6-j symbol may be derived by considering different ways of achieving the same ultimate recoupling of three angular momenta. Clearly, (j 1 j 2 )j 12 j 3 j j 2 (j 3 j 1 )j 13 j = (j 1 j 2 )j 12 j 3 j j 1 (j 2 j 3 )j 23 j j 1 (j 2 j 3 )j 23 j j 2 (j 3 j 1 )j 13 j. j 23 The transformation coefficients on both sides can be reduced to standard form by changing the order of coupling of pairs of angular momenta, with the appropriate phase factors, and rewritten in terms of 6-j symbols to yield ( ) j 12+j 13+j 23 j 1 j 2 j 12 j 1 j 3 j 13 (2j ) j 23 j 3 j j 23 j 2 j j 23 = j 1 j 3 j 13 j j 2 j 12. (3.11) A somewhat more complex sum rule is obtained by considering the recoupling of four angular momenta, three at a time, in different ways. In an obvious notation, (j 1 j 2 )j 12 j 3 j 123 j 4 j (j 2 j 3 )j 23 j 1 j 123 j 4 j (j 2 j 3 )j 23 (j 1 j 4 )j 14 j; or (j 1 j 2 )j 12 j 3 j 123 j 4 j (j 1 j 2 )j 12 j 4 j 124 j 3 j (j 1 j 4 )j 14 j 2 j 124 j 3 j (j 1 j 4 )j 14 (j 2 j 3 )j 23 j. The resulting equality can be converted into the Biedenharn-Elliott sum rule for 6-j symbols, ( ) j 1+j 2+j 3+j 4+j 12+j 23+j 14+j 123+j j 124+j 1 j 2 j 12 j 12 j 3 j 123 (2j ) j 124 j 124 j 4 j 14 j j 4 j 124 j 2 j 3 j 23 j j 14 j 124 = j 1 j 23 j 123 j 1 j 23 j 123 j j 4 j 14 j 3 j 12 j 2. (3.12) 18

19 The value of the 6-j symbol in the special case where one of its arguments is zero is easily determined by substitution in the equation relating the 6-j symbol to the Clebsch-Gordan coefficients, or by observing that (j 1 j 2 )j 12 0j j 1 (j 2 0)j 23 j = δ j12 jδ j2 j 23. Either procedure produces the result j 1 j 2 j j 2 j 1 0 = ( )j 1+j 2 +j δ j1 j 1 δ j 2 j 2 / (2j 1 + 1)(2j 2 + 1). (3.13) The symmetry properties of the 6-j symbol can then be used to evaluate a 6-j symbol any of whose arguments is zero. Inserting this special value of the 6-j symbol into the orthogonality relation or any of the sum rules for these symbols, by setting one of the free angular momenta (not an index of summation) equal to zero, will generally produce a trivial identity, but in a few cases it produces further useful sum rules. Setting j 12 = 0 in the orthogonality relation leads to ( ) j 1+j 2+j 12 j 1 j 2 j 12 (2j ) j 12 j 2 j 1 j = (2j 1 + 1)(2j 2 + 1)δ j0. (3.14) Similarly, setting j 13 = 0 in the first sum rule above produces j 1 j 2 j 12 (2j + 1) j j 1 j 2 j = ( )2(j 1+j 2 ), which can be rewritten in the form j ( ) 2j 1 j 2 j 12 (2j + 1) = 1. (3.15) j j 1 j 2 j Setting an argument equal to zero in the Biedenharn-Elliott sum rule produces no new results. For the case of four coupled angular momenta, the uncoupled basis j 1 m 1 j 2 m 2 j 3 m 3 j 4 m 4 is related to the fully coupled basis j 1 j 2 j 3 j 4 j j jm (where j and j are intermediate couplings required to complete a full set of eight mutually commuting operators) by a unitary transformation. The number of possible sets of fully coupled basis functions is now quite 19

20 large (illustrative examples are j = j 12, j = j 123 ; j = j 13, j = j 134 ; j = j 12, j = j 34 ), but they fall into two classes bases where two angular momenta are coupled together, the result coupled to a third and that result to the fourth angular momentum, or bases where two distinct pairs of angular momenta are each coupled together and then their resultants are coupled to produce the total angular momentum. Any two such bases are related by a unitary transformation. The first type is of considerably less interest than the second and can be handled by successive uses of the 6-j symbol. The second type is sufficiently useful to have warranted the introduction of a new recoupling coefficient. The prototype transformation coefficient for four angular momenta coupled pairwise is (j 1 j 2 )j 12 (j 3 j 4 )j 34 j (j 1 j 3 )j 13 (j 2 j 4 )j 24 j, where again the overlap between states from different bases is diagonal in j and m and independent of m, which is dropped from the transformation bracket. (The overlap is also, of course, diagonal in the common quantum numbers j 1, j 2, j 3, j 4.) This can be written in terms of Clebsch-Gordan coefficients as j 1 m 1 j 2 m 2 j 12 m 12 j 3 m 3 j 4 m 4 j 34 m 34 j 12 m 12 j 34 m 34 jm m 1 m 2 m 3 m 4 m 12 m 34 m 13 m 24 j 1 m 1 j 3 m 3 j 13 m 13 j 2 m 2 j 4 m 4 j 24 m 24 j 13 m 13 j 24 m 24 jm, and is hence real. Once again it turns out to be convenient to introduce a slightly different recoupling coefficient, the Wigner 9-j symbol, defined by (j 1 j 2 )j 12 (j 3 j 4 )j 34 j (j 1 j 3 )j 13 (j 2 j 4 )j 24 j = j 1 j 2 j 12 (2j )(2j )(2j )(2j ) j 3 j 4 j 34. (3.16) j 13 j 24 j Converting the Clebsch-Gordan coefficients to 3-j symbols and exploiting the m- independence of the recoupling coefficient, it is found that j 1 j 2 j 12 j 3 j 4 j 34 = j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j j 13 j 24 j all m s m 1 m 2 m 12 m 3 m 4 m 34 m 13 m 24 m j 1 j 3 j 13 j 2 j 4 j 24 j 12 j 34 j (3.17) m 1 m 3 m 13 m 2 m 4 m 24 m 12 m 34 m where each row and each column of the 9-j symbol is represented by a 3-j symbol and all 20

21 nine projection quantum numbers m are summed over. The result, independent of projection quantum numbers, may be regarded as a rotational invariant. The selection rules for the 9-j symbol are now trivial each row and each column of the symbol contributes a triangle inequality which must be satisfied if the symbol is not to vanish. The symmetries of the 9-j symbol are also easily read off: the interchange of a pair of rows or of a pair of columns produces a phase factor ( ) j 1+j 2 +j 3 +j 4 +j 12 +j 34 +j 13 +j 24 +j, while transposition of the symbol (writing the rows as columns and the columns as rows by reflecting in the main diagonal) simply reorders the six 3-j symbols inside the summation and so leaves the symbol unchanged in value. The former symmetry introduces a further selection rule if any two columns or any two rows of the symbol are identical, then it vanishes unless the sum of its nine arguments is even (or, equivalently, unless the sum of the angular momenta in the remaining column or row is even). As before, the unitarity of the transformation imposes orthogonality conditions on the 9-j symbols, in the form j 1 j 2 j 12 j 1 j 2 j 12 (2j )(2j ) j 3 j 4 j 34 j 3 j 4 j = δ 34 j12 j δ 12 j 34 j /(2j )(2j ). j 13 j 24 j 13 j 24 j j 13 j 24 j (3.18) The 9-j symbol can be expressed in terms of 6-j symbols by recoupling four angular momenta in stages, with three angular momenta being recoupled at each stage, as follows: (j 1 j 2 )j 12 (j 3 j 4 )j 34 j (j 1 j 3 )j 13 (j 2 j 4 )j 24 j = j 123 (j 1 j 2 )j 12 (j 3 j 4 )j 34 j (j 1 j 2 )j 12 j 3 j 123 j 4 j (j 1 j 2 )j 12 j 3 j 123 j 4 j (j 1 j 3 )j 13 j 2 j 123 j 4 j (j 1 j 3 )j 13 j 2 j 123 j 4 j (j 1 j 3 )j 13 (j 2 j 4 )j 24 j from which it follows that j 1 j 2 j 12 j 3 j 4 j 34 = j ( ) 2j j 2 j 12 j 3 j 4 j 34 j 13 j 24 j (2j ) j 123 j j 13 j 24 j 123 j 3 j 13 j j 12 j 123 j 4 j 123 j 2. (3.19) As in the case of the expression for the 6-j symbol in terms of Clebsch-Gordan coefficients, this result may be manipulated, using the orthogonality of the 6-j symbols, to express a product of two 6-j symbols as a sum of products of a 6-j and a 9-j symbol. Successive recouplings of four angular momenta can be used to obtain a sum rule expressing a single 21

22 9-j symbol as a sum of products of pairs of 9-j symbols. The value of the 9-j symbol when one of its arguments vanishes is also easily obtained from the expression in terms of 6-j symbols. Given the symmetries of the 9-j symbol, it is sufficient to evaluate j 1 j 2 j 12 j 3 j 4 j 34 = ( ) j 2+j 12 +j 3 +j 13 j 1 j 2 j 12 δ j12 j 34 δ j13 j 24 j j 13 j j 3 j 13 / (2j )(2j ). (3.20) A useful special case arises when a whole row or column of arguments vanishes. This is covered by the specific result j 1 j 1 0 j 2 j 2 0 = δ j1 j δ 1 j 2 j δ 2 j 3 j (2j / )(2j 2 + 1)(2j 3 + 1), (3.21) j 3 j 3 0 where j 1, j 2 and j 3 satisfy the triangle relation. IV. SPHERICAL TENSOR OPERATORS Now that the quantum states of a system have been characterized in terms of their angular momentum properties, it becomes of interest to investigate the effect on these properties of various operators which act in the space of states. Consider, in the simplest case, the position and momentum operators of a single spinless particle with orbital angular momentum operator L = ( r p)/ h. Using the standard commutation relation [r α, p β ] = i hδ αβ, where the indices α, β represent the cartesian components x, y, z, it is easily established that the commutator [L α, r β ] is a simple linear combination of the components of r, while [L α, p β ] is the same linear combination of the corresponding components of p. It is possible to define specific linear combinations r m and p m of the components of r and p respectively, with m = 1, 0, +1, such that where v represents either r or p and where [L z, v m ] = mv m (4.1) [L ±, v m ] = α (±) 1m v m±1 (4.2) v 0 = v z (4.3) v ±1 = 1 (v x ± iv y ). 2 (4.4) 22

23 These eigenoperators of L have properties reminiscent of the defining properties of a set of states of angular momentum 1, projection m, with the commutator of L with the operators v m playing the role of the action of the operator L on the states 1m. This property of the operators r and p can be generalized and is extremely useful in all applications of the angular momentum algebra. Before going on to the general case, consider the set of operators r α p β, the nine possible products of the cartesian components (α, β = x, y, z) of r and p. Once again, the commutators of L with these operators produce only linear combinations of operators in the same set. This is guaranteed by the fact that r and p have this property, while commutators satisfy the identity [A, BC] = [A, B]C + B[A, C]. The operator products r α p β fall naturally into three subsets the combination r p (the sum of the diagonal elements of the set, called the trace), the vector product r p (the three antisymmetric combinations r α p β r β p α ) and the five independent symmetric combinations r α p β + r β p α 2 3 δ αβ r p, with vanishing trace. Each product r α p β can be written as a sum of terms from the three subsets. It is then found that each of these subsets is closed under commutation with L, i.e. the commutator of any component of L with a member of one of the subsets is a linear combination of members of the same subset. The scalar product r p commutes with all the components of L; the vector product r p has the same commutation relations with L as the individual vectors r and p; while specific linear combinations ( r p) m of the elements of the symmetric subset can be found for which [L z, ( r p) m ] = m( r p) m (4.5) [L ±, ( r p) m ] = α (±) 2m ( r p) m±1. (4.6) These results generalize to other operators than r, p and their products and to more general angular momentum operators than L. Given any set of operators which is closed under commutation with the components of the angular momentum j, it can always be broken up into subsets, each of which is itself closed under commutation with j but cannot be broken up into smaller subsets with the same property of closure. Any of the original operators can be written as a linear combination of operators from these subsets. Within each such irreducible subset, specific linear combinations can be chosen so that their commutation relations with j have the standard form encountered twice above, analogous to 23

24 that of sets of states jm acted upon by the operator j. Further it can be demonstrated that any arbitrary operator can be written as a linear combination of operators belonging to such standard sets. These general statements are systematized in the definition of an (irreducible) spherical tensor operator. An (irreducible) spherical tensor operator T (j) m is one of a set of 2j + 1 operators, corresponding to different values of m = j, j + 1,..., j 1, j, which satisfy The spherical tensor operator T (j) m [j z, T m (j) (j) ] = mt m (4.7) [ ] j±, T m (j) (±) = α jm T m±1. (j) (4.8) is said to be of rank j, projection m. This definition essentially generalizes the notion of angular momentum of states to include that of angular momentum of operators. It is now possible to investigate the effect on the angular momentum of a state of the action of a spherical tensor operator. Consider the state produced by operating with T (j) m on j m. It is still an eigenstate of j z, since j z T (j) m j m = [j z, T (j) m ] j m + T (j) m j z j m = mt (j) m j m + T (j) m m j m = (m + m )T (j) m j m, but its eigenvalue has become m + m. It is tedious but straightforward to check that T (j) m j m is not, however, an eigenstate of j 2. But it is essentially a product of two objects of well-defined angular momenta j and j, with well-defined projection quantum numbers m and m. The analogy with the product state j 1 m 1 j 2 m 2 suggests that an appropriate linear combination of such products using, of course, the Clebsch-Gordan coefficients may have well-defined angular momentum. Consider, therefore, the state JM jj = mm jmj m JM T (j) m j m. Because of the Clebsch-Gordan selection rule M = m + m, this is, in fact, an eigenstate of 24

25 j z with eigenvalue M, as suggested by the notation. Now, j ± JM jj = mm jmj m JM {[j ±, T (j) m ] + T (j) m j ± } j m = jmj m JM {α (±) jm T m±1 j (j) m + α (±) mm (j) j m T m j, m ± 1 } = mm [α (±) j,m 1 j, m 1, j m JM + α (±) j,m 1 jmj, m 1 JM ]T (j) m j m = mm α (±) JM jmj m J, M ± 1 T (j) m j m = α (±) JM J, M ± 1 jj where use has been made of the Clebsch-Gordan coefficient recursion relations and of the restrictions on the ranges of the projection quantum numbers. But this result is precisely what is required to establish JM jj as an eigenstate of j 2 with eigenvalue J(J + 1), thus fully justifying the notation used. The Clebsch-Gordan coefficients thus couple the rank of a spherical tensor operator to the angular momentum of the state on which it acts so that the state so produced is a welldefined angular momentum eigenstate. The orthogonality of the Clebsch-Gordan coefficients can now be used to write T (j) m j m = JM jmj m JM JM jj, which can be used to evaluate the matrix element j m T (j) m j m = JM jmj m JM j m JM jj = JM jmj m JM Θ j jj δ j Jδ m M = jmj m j m Θ j jj, where Θ j jj, the overlap between the states j m and JM jj, is diagonal in j (J) and in m (M) and independent of M. Therefore the dependence of the matrix element on the projection quantum numbers is contained entirely in the Clebsch-Gordan coefficient and is independent of the detailed dynamics of the states and of the operator involved. This very powerful and extremely useful result is known as the Wigner-Eckart theorem. It should be emphasized that all that is required for the theorem to hold is for the two states concerned to be eigenstates of the same angular momentum operator relative to which the operator involved is a spherical tensor operator. Any such matrix element can then be regarded as 25